For \(n=1\)  the expansion (REF ) reduces to (REF ) which is a simple check. A more sophisticated check is provided by \(Q_3, Q_5\)  and \(Q_7\)  which are known explicitly in terms of the Virasoro algebra generators [1]}
\(Q_3&=&\left(L_0^2-{c+2\over 12}L_0+{c(5c+22)\over 2990}\right)+\tilde{Q}_3,\\ \nonumber \tilde{Q}_3&=&2\sum _{k=1}^\infty L_{-k}L_k,\\Q_5&=&\left(L_0^3- \frac{c+4}{8} L_0^2 + \frac{(c+2)(3c + 20)}{576}L_0 - \frac{c(3c + 14)(7c + 68)}{290304}\right)+ \tilde{Q}_5, \\\tilde{Q}_5 &=& \sum _{k, l = 0}^{\infty } L_{-k-l} L_k L_l + 2\sum _{k = 1, l = 0}^{\infty } L_{-k} L_{k-l} L_l + \sum _{k, l = 1}^{\infty } L_{-k} L_{-l} L_{k+l} \, + \nonumber \\ &+& \sum _{n=1}^{\infty } \left(\frac{c+2}{6}m^2 -\frac{c}{4} - 1 \right) L_{-n}L_n - L_0^3,\) 
